Q60 / 120 · VITEEE 2011 · Mathematics

If $a, b, c$ are three non-zero vectors such that $\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=0$ and $m=\boldsymbol{a} \cdot \boldsymbol{b}+\boldsymbol{b} \cdot \boldsymbol{c}+\boldsymbol{c} \cdot \boldsymbol{a}$, then

  • Correct answer
    A
    $m < 0$
  • B
    $m>0$
  • C
    $m=0$
  • D
    $m=3$

Solution

$$ \begin{array}{l} \text { Here, } \mathbf{a}+\mathbf{b}+\mathbf{c}=0 \\ \Rightarrow|\mathbf{a}+\mathbf{b}+\mathbf{c}|^{2}=0 \\ \Rightarrow|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+\mid \mathbf{c}^{2}+2\{\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}\}=0 \\ \Rightarrow \mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a} \\ =-\frac{1}{2}\left\{|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+|\mathbf{c}|^{2}\right\} < 0 \\ \Rightarrow \mathrm{m} < 0 \end{array} $$

Hence, the correct answer is option: 'A'.

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